Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $r = \dfrac{2p^2 - 12p + 10}{3p^3 + 18p^2 - 48p} \times \dfrac{-4p^2 - 32p}{-5p + 25} $
Explanation: First factor out any common factors. $r = \dfrac{2(p^2 - 6p + 5)}{3p(p^2 + 6p - 16)} \times \dfrac{-4p(p + 8)}{-5(p - 5)} $ Then factor the quadratic expressions. $r = \dfrac {2(p - 5)(p - 1)} {3p(p + 8)(p - 2)} \times \dfrac {-4p(p + 8)} {-5(p - 5)} $ Then multiply the two numerators and multiply the two denominators. $r = \dfrac { 2(p - 5)(p - 1) \times -4p(p + 8)} { 3p(p + 8)(p - 2) \times -5(p - 5)} $ $r = \dfrac {-8p(p - 5)(p - 1)(p + 8)} {-15p(p + 8)(p - 2)(p - 5)} $ Notice that $(p + 8)$ and $(p - 5)$ appear in both the numerator and denominator so we can cancel them. $r = \dfrac {-8p(p - 5)(p - 1)\cancel{(p + 8)}} {-15p\cancel{(p + 8)}(p - 2)(p - 5)} $ We are dividing by $p + 8$ , so $p + 8 \neq 0$ Therefore, $p \neq -8$ $r = \dfrac {-8p\cancel{(p - 5)}(p - 1)\cancel{(p + 8)}} {-15p\cancel{(p + 8)}(p - 2)\cancel{(p - 5)}} $ We are dividing by $p - 5$ , so $p - 5 \neq 0$ Therefore, $p \neq 5$ $r = \dfrac {-8p(p - 1)} {-15p(p - 2)} $ $ r = \dfrac{8(p - 1)}{15(p - 2)}; p \neq -8; p \neq 5 $